A piece in The Chronicle of Higher Education says no. 
Essentially, the author does not believe that digitization will make physical libraries a thing of the past:
Everywhere in the country librarians report that they have never had so many patrons. At Harvard, our reading rooms are full. The 85 branch libraries of the New York Public Library system are crammed with people. The libraries supply books, videos, and other material as always, but they also are fulfilling new functions: access to information for small businesses, help with homework and afterschool activities for children, and employment information for job seekers (the disappearance of want ads in printed newspapers makes the library’s online services crucial for the unemployed).
But how could we test this claim? So long as libraries–such as the New York system’s branch libraries–are funded through public means, they could never become obsolete. Massive subsidies maintain their existence.
If physical libraries are fulfilling a legitimate need in the information age, they would thrive even without public funding.
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